Where in the 5e rules is "supernatural" defined as "not actually magic"?
3e had that as a keyword on types of abilities - I don't recall that being the case in 5e. So, this sounds like your personal interpretation, rather than a fact of the game for everyone.
No, before you jump the gun, the distinction is contained in the rules.
On Page 20:
Is the breath weapon of a dragon magical? If you cast antimagic field, don armor of invulnerability, or use another feature of the game that protects against magical or nonmagical effects, you might ask yourself, “Will this protect me against a dragon’s breath?” The breath weapon of a typical dragon isn’t considered magical, so antimagic field won’t help you but armor of invulnerability will.
You might be thinking, “Dragons seem pretty magical to me.” And yes, they are extraordinary! Their description even says they’re magical. But our game makes a distinction between two types of magic:
• the background magic that is part of the D&D multiverse’s physics and the physiology of many D&D creatures
• the concentrated magical energy that is contained in a magic item or channeled to create a spell or other focused magical effect
In D&D, the first type of magic is part of nature. It is no more dispellable than the wind. A monster like a dragon exists because of that magic-enhanced nature. The second type of magic is what the rules are concerned about. When a rule refers to something being magical, it’s referring to that second type.
Determining whether a game feature is magical is straightforward. Ask yourself these questions about the feature:
• Is it a magic item?
• Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description?
• Is it a spell attack?
• Is it fueled by the use of spell slots?
• Does its description say it’s magical?
If your answer to any of those questions is yes, the feature is magical.
Let’s look at a white dragon’s Cold Breath and ask ourselves those questions. First, Cold Breath isn’t a magic item. Second, its description mentions no spell. Third, it’s not a spell attack. Fourth, the word “magical” appears nowhere in its description. Our conclusion: Cold Breath is not considered a magical game effect, even though we know that dragons are amazing, supernatural beings.
Dragons (and their breath weapons, and ability to fly) are supernatural things, and part of the background 'supernatural physics' of the world (like constructs moving about, or ghosts existing, or beholders hovering). But the game rules makes a distinction between 'supernatural' and overtly 'magical'.
Applying the above rules to Pact Weapon class feature:
• Is it a magic item? No, it only counts as one for the sole purpose of resistance and immunity - not for any other purpose.
• Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description? - No
• Is it a spell attack? -No
• Is it fueled by the use of spell slots? -No
• Does its description say it’s magical? -No
Neither the ability of the Warlock to summon the pact weapon, nor the weapon itself is magical. Supernatural for sure, but not 'magical'.
Note the difference between the Warlocks 'Pact Weapon' class feature and the Eldritch Knights 'Weapon Bond' class feature. The latter EK ability is magical (because the feature states it is magical in the description of the class feature), and the Bond does not work (and is suppressed) in an AMF.
Pact Weapon on the other hand, is not magical, and is not suppressed.